# New Perturbation Bounds for the Unitary Polar Factor

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/CSD-94-852.pdf

Let A be an m x n ( m >= n) complex matrix. It is known that there is a unique polar decomposition A = QH, where Q* Q = I, the n x n identity matrix, and H is positive definite, provided A has full column rank. This note addresses the following question: how much may Q change if A is perturbed? For the square case m = n our bound, which is valid for any unitarily invariant norm, is sharper and simpler than Mathias's ( SIAM J. Matrix Anal. Appl., 14 (1993), 588-597.). For the non-square case, we also establish a bound for unitarily invariant norm, which has not been done in literature.

BibTeX citation:

```@techreport{Li:CSD-94-852,
Author = {Li, Ren-Cang},
Title = {New Perturbation Bounds for the Unitary Polar Factor},
Institution = {EECS Department, University of California, Berkeley},
Year = {1994},
Month = {Dec},
URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5883.html},
Number = {UCB/CSD-94-852},
Abstract = {Let <i>A</i> be an <i>m</i> x <i>n</i> (<i>m</i> >= <i>n</i>) complex matrix. It is known that there is a unique polar decomposition <i>A</i> = <i>QH</i>, where <i>Q</i>*<i>Q</i> = <i>I</i>, the <i>n</i> x <i>n</i> identity matrix, and <i>H</i> is positive definite, provided <i>A</i> has full column rank. This note addresses the following question: how much may <i>Q</i> change if <i>A</i> is perturbed? For the square case <i>m</i> = <i>n</i> our bound, which is valid for any unitarily invariant norm, is sharper and simpler than Mathias's (<i>SIAM J. Matrix Anal. Appl., <b>14</b> (1993), 588-597.</i>). For the non-square case, we also establish a bound for unitarily invariant norm, which has not been done in literature.}
}
```

EndNote citation:

```%0 Report
%A Li, Ren-Cang
%T New Perturbation Bounds for the Unitary Polar Factor
%I EECS Department, University of California, Berkeley
%D 1994
%@ UCB/CSD-94-852
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5883.html
%F Li:CSD-94-852
```